Recently, I was reading a paper about the rigidity of negatively curved cone surfaces written by S. Hersonsky and F. Paulin. The authors said that a negatively curved cone surface is locally CAT(-1). But I don't know why.
A negatively curved cone surface is a surface $M$ endowed with a negatively curved cone metric, i.e. a smooth negatively curved Riemannian metric on $M-P$, where $P$ is a discrete subset of points on $M$, such that the completion of $M-P$ is $M$, and such that the cone angle at each singularity is greater than $2\pi$.